## Regularity of solutions to an integral equation associated with Bessel potential.(English)Zbl 1237.45002

The authors study the differential equation in $$\mathbb{R}^n$$ $(I-\Delta)^{\frac{\alpha}{2}} u = u^\beta,\quad \alpha >0, \;\beta > 1.$ They prove that if $$u$$ is a positive solution of this equation in the space $$L^q (\mathbb{R}^n), \;q > \max (\beta, \frac{n(\beta - 1)}{\alpha}),$$ then $$u$$ is a bounded and Lipschitz continuous function.
There is a survey of previous results in the article.

### MSC:

 45E10 Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type) 35J60 Nonlinear elliptic equations 45M20 Positive solutions of integral equations
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